Simultaneous linear equations

A table of values lists two groups of numbers: this table lists the "input values" the top and the "output values" on the bottom. The numbers which are above and below each other are related to each other by the equation!

$x$ is always the input value for an equation, and $y$ is always the output value. For the first question, we need to find the output value which corresponds to the input value of $x = 8$: to find the corresponding output value, we substitute $x = 8$ into the equation.

The table of values shows the input-output pair $x = 8$ and $y = 34$:

The output value for the input of $x = 8$ is 34.

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of $y = 10$ is $2$.

A table of values lists two groups of numbers: this table lists the "input values" the top and the "output values" on the bottom. The numbers which are above and below each other are related to each other by the equation!

$x$ is always the input value for an equation, and $y$ is always the output value. For the first question, we need to find the output value which corresponds to the input value of $x = 7$: to find the corresponding output value, we substitute $x = 7$ into the equation.

The table of values shows the input-output pair $x = 7$ and $y = 9$:

The output value for the input of $x = 7$ is 9.

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of $y = 11$ is $9$.

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between $x$ and $y$.

$x$ is always the input value for an equation, and $y$ is always the output value. For the first question, we need to find the output value which corresponds to the input value of $x = 9$: to find the corresponding output value, we substitute $x = 9$ into the equation.

The table of values shows the input-output pair $x = 9$ and $y = 14$:

The output value for the input of $x = 9$ is 14.

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of $y = 15$ is $10$.

You need to find the coordinates of the point where the lines intersect.

This question is about two equations. We need to find the values of $x$ and $y$ which solve the equations 'simultaneously'. This means we want one pair of $x$ and $y$ values which solves both equations.

There are different methods to solve equations simultaneously. In this case we can use the graph: the solutions we want are the coordinates where the lines intersect. The intersection point is very special - it is the only point shared by both lines. And the coordinates of that point are the only numbers which solve both equations! In this case the lines intersect at the point $\left(2; -1\right)$.

We can prove that this is correct by substituting the values into each of the equations:

and

Perfect - the values $x = 2$ and $y = -1$ solve the equations simultaneously.

The correct answers are $x = 2$ and $y = -1$.

You need to find the coordinates of the point where the lines intersect.

This question is about two equations. We need to find the values of $x$ and $y$ which solve the equations 'simultaneously'. This means we want one pair of $x$ and $y$ values which solves both equations.

There are different methods to solve equations simultaneously. In this case we can use the graph: the solutions we want are the coordinates where the lines intersect. The intersection point is very special - it is the only point shared by both lines. And the coordinates of that point are the only numbers which solve both equations! In this case the lines intersect at the point $\left(1; 5\right)$.

We can prove that this is correct by substituting the values into each of the equations:

and

Perfect - the values $x = 1$ and $y = 5$ solve the equations simultaneously.

The correct answers are $x = 1$ and $y = 5$.

You need to find the coordinates of the point where the lines intersect.

This question is about two equations. We need to find the values of $x$ and $y$ which solve the equations 'simultaneously'. This means we want one pair of $x$ and $y$ values which solves both equations.

There are different methods to solve equations simultaneously. In this case we can use the graph: the solutions we want are the coordinates where the lines intersect. The intersection point is very special - it is the only point shared by both lines. And the coordinates of that point are the only numbers which solve both equations! In this case the lines intersect at the point $\left(-4; 3\right)$.

We can prove that this is correct by substituting the values into each of the equations:

and

Perfect - the values $x = -4$ and $y = 3$ solve the equations simultaneously.

The correct answers are $x = -4$ and $y = 3$.

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between $x$ and $y$.

$x$ is always the input value for an equation, and $y$ is always the output value. For the first question, we need to find the output value which corresponds to the input value of $x = 2$: to find the corresponding output value, we substitute $x = 2$ into the equation.

The table of values shows the input-output pair $x = 2$ and $y = -1$:

The output value for the input of $x = 2$ is -1.

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of $y = 3$ is 4.

A table of values lists two groups of numbers: this table lists the "input values" the top and the "output values" on the bottom. The numbers which are above and below each other are related to each other by the equation!

$x$ is always the input value for an equation, and $y$ is always the output value. For the first question, we need to find the output value which corresponds to the input value of $x = 2$: to find the corresponding output value, we substitute $x = 2$ into the equation.

The table of values shows the input-output pair $x = 2$ and $y = 3$:

The output value for the input of $x = 2$ is 3.

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of $y = 19$ is 10.

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between $x$ and $y$.

$x$ is always the input value for an equation, and $y$ is always the output value. For the first question, we need to find the output value which corresponds to the input value of $x = 6$: to find the corresponding output value, we substitute $x = 6$ into the equation.

However, this equation does not even have an $x$ in it - that is a bit weird! What this means is that the output is not related to the input: the output is always -4, no matter what the input is. (You can see that easily on the table of values: all of the output numbers are the same!) It is like a student who must wake up at 06:00 in the morning for school: if he slept early or stayed up late, it does not matter - he still must get up at 06:00. When he wakes does not depend on the time when he sleeps, just like the output for this equation does not depend on the input.

The table of values shows the input-output pair $x = 6$ and $y = -4$:

The output value for the input of $x = 6$ is -4.

This question has multiple answers! Since the output value is always -4, all of the input values lead to the output value of $y = -4$. Therefore, you can give any one of these answers for the second question: 1, 3, 5 or 6.

You should compare the coefficients in the two equations, and look for like terms: if you collect those terms, which terms will cancel?

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is elimination. The elimination method works by cancelling like terms which the two equations share. We need to figure out which of the terms in the equations will cancel, and why that cancellation happens.

In these two questions, there are two $x$-terms and two $y$-terms:

The goal of elimination is to cancel out like terms by adding (or subtracting) the equations. That can be either the $x$-terms or the $y$-terms.

For the equations in this question, notice that the coefficients of $y$ are equal and opposite. So if we add them together, they will cancel. So we will add the equations together to make that cancellation happen, like this:

The $y$-terms cancel each other out - that is what 'elimination' means! We get the $0y$ because the $y$-coefficients are equal and opposite: $5y + \left(-5y\right)$ is equal to zero. So the elimination depends on which of the coefficients cancel out.

The correct answer choices are: We will eliminate the $y$-terms because the $y$-coefficients will cancel.

You should compare the coefficients in the two equations, and look for like terms: if you collect those terms, which terms will cancel?

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is elimination. The elimination method works by cancelling like terms which the two equations share. We need to figure out which of the terms in the equations will cancel, and why that cancellation happens.

In these two questions, there are two $x$-terms and two $y$-terms:

The goal of elimination is to cancel out like terms by adding (or subtracting) the equations. That can be either the $x$-terms or the $y$-terms.

For the equations in this question, notice that the coefficients of $x$ are equal and opposite. So if we add them together, they will cancel. So we will add the equations together to make that cancellation happen, like this:

The $x$-terms cancel each other out - that is what 'elimination' means! We get the $0x$ because the $x$-coefficients are equal and opposite: $5x + \left(-5x\right)$ is equal to zero. So the elimination depends on which of the coefficients cancel out.

The correct answer choices are: We will eliminate the $x$-terms because the $x$-coefficients will cancel.

You should compare the coefficients in the two equations, and look for like terms: if you collect those terms, which terms will cancel?

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is elimination. The elimination method works by cancelling like terms which the two equations share. We need to figure out which of the terms in the equations will cancel, and why that cancellation happens.

In these two questions, there are two $x$-terms and two $y$-terms:

The goal of elimination is to cancel out like terms by adding (or subtracting) the equations. That can be either the $x$-terms or the $y$-terms.

For the equations in this question, notice that the coefficients of $y$ are equal and opposite. So if we add them together, they will cancel. So we will add the equations together to make that cancellation happen, like this:

The $y$-terms cancel each other out - that is what 'elimination' means! We get the $0y$ because the $y$-coefficients are equal and opposite: $-y + \left(y\right)$ is equal to zero. So the elimination depends on which of the coefficients cancel out.

The correct answer choices are: We will eliminate the $y$-terms because the $y$-coefficients will cancel.

The first thing to do is compare the coefficients of the terms. We can see that the coefficient of the $g$-term in the first equation is a multiple of the second $g$-term: the $\text{2}$ is $\text{4}$ times as much as $\text{8}$. If we multiply the entire second equation by this factor of $\text{4}$ we will be able to cancel the $g$-terms between the equations:

Now eliminate the $g$-terms. Since the terms have the same signs, we must subtract one equation from the other to cancel those terms.

Now we can solve for $f$:

Finally, use the value of $f$ to find the value of $g$. Remember that we can use either of the equations to do this, we can pick the easier equation. The equations are pretty much the same (they have the same number of negative signs, for example, and neither has any fractions). So we will just pick the first equation.

The answers to the pair of equations are $f = -2$ and $g = -1$.

The first thing to do is compare the coefficients of the terms. We can see that the coefficient of the $b$-term in the first equation is a multiple of the second $b$-term: the $\text{2}$ is $\text{4}$ times as much as $\text{8}$. If we multiply the entire second equation by this factor of $\text{4}$ we will be able to cancel the $b$-terms between the equations:

Now eliminate the $b$-terms. Since the terms have opposite signs, we must add one equation to the other to cancel those terms.

Now we can solve for $a$:

Finally, use the value of $a$ to find the value of $b$. Remember that we can use either of the equations to do this, we can pick the easier equation. The second equation looks like a better choice because it has fewer negative signs than the first equation (in fact it does not have any!).

The answers to the pair of equations are $a = 8$ and $b = 7$.

The first thing to do is compare the coefficients of the terms. We can see that the coefficient of the $j$-term in the first equation is a multiple of the second $j$-term: the $\text{6}$ is $\text{2}$ times as much as $\text{12}$. If we multiply the entire second equation by this factor of $\text{2}$ we will be able to cancel the $j$-terms between the equations:

Now eliminate the $j$-terms. Since the terms have the same signs, we must subtract one equation from the other to cancel those terms.

Now we can solve for $k$:

Finally, use the value of $k$ to find the value of $j$. Remember that we can use either of the equations to do this, we can pick the easier equation. In this case the first equation has fewer negatives than the second equation - in fact it has none. So we will use the first equation.

The answers to the pair of equations are $j = 4$ and $k = -7$.

We need to solve the equations in this equation using elimination. (Substitution will also work, but the question says that we should use elimination.) As always for elimination, we need to use the two equations to cancel some of the terms.

By comparing the coefficients of the terms, we can see that the second $x$-term has a coefficient which is a multiple of the first $x$-term. In fact, $\text{3}$ is $\text{3}$ times as much as $\text{1}$. We can make both coefficients of $x$ the same by using this factor of $\text{3}$ to multiply the entire first equation:

Now we can eliminate the $x$-terms by subtracting one equation from the other. Then we can solve for $y$.

Now we can solve for $y$:

The last step is to substitute $y$ back into either of the equations so that we can find $x$. Here we will use the first equation:

The answers are $x = 10$ and $y = 4$.

We need to solve the equations in this equation using elimination. (Substitution will also work, but the question says that we should use elimination.) As always for elimination, we need to use the two equations to cancel some of the terms.

By comparing the coefficients of the terms, we can see that the second $x$-term has a coefficient which is a multiple of the first $x$-term. In fact, $\text{12}$ is $\text{3}$ times as much as $\text{4}$. We can make both coefficients of $x$ the same by using this factor of $\text{3}$ to multiply the entire first equation:

Now we can eliminate the $x$-terms by subtracting one equation from the other. Then we can solve for $y$.

Now we can solve for $y$:

The last step is to substitute $y$ back into either of the equations so that we can find $x$. Here we will use the first equation:

The answers are $x = -5$ and $y = -7$.

We need to solve the equations in this equation using elimination. (Substitution will also work, but the question says that we should use elimination.) As always for elimination, we need to use the two equations to cancel some of the terms.

By comparing the coefficients of the terms, we can see that the second $y$-term has a coefficient which is a multiple of the first $y$-term. In fact, $\text{5}$ is $\text{5}$ times as much as $\text{1}$. We can make both coefficients of $y$ the same by using this factor of $\text{5}$ to multiply the entire first equation:

Now we can eliminate the $y$-terms by subtracting one equation from the other. Then we can solve for $x$.

Now we can solve for $x$:

The last step is to substitute $y$ back into either of the equations so that we can find $x$. Here we will use the first equation:

The answers are $x = -1$ and $y = -3$.

Remember that the goal of elimination is to cancel either the $k$-terms or the $p$-terms. So you need to change one of the coefficients in the first equations so that you can cancel terms.

For this question we need to multiply the first equation by a number so that we can solve the equations by elimination. In the original equations, none of the coefficients were ready to cancel (because none of them are equal). The second equation was already multiplied by $\text{3}$. We need to change one of the coefficients in the first equation so that we can cancel terms.

Specifically, we need to multiply $\text{4}$ to get $-\text{9}$ so we can cancel the $k$-terms, or multiply the $\text{3}$ to get $-\text{15}$ so we can cancel the $p$-terms.

The numbers in these equations point us toward cancelling the $p$-terms. This is because $-\text{15}$ is a multiple of $\text{3}$. So if we multiply the first equation by $\text{5}$ the the coefficients will be equal and opposite.

The coefficients we want to cancel are equal and opposite. So eliminating the terms requires adding both sides of the equation:

The correct answer can be either of these numbers: $-\text{5}$ or $\text{5}$.

Remember that the goal of elimination is to cancel either the $a$-terms or the $b$-terms. So you need to change one of the coefficients in the first equations so that you can cancel terms.

For this question we need to multiply the first equation by a number so that we can solve the equations by elimination. In the original equations, none of the coefficients were ready to cancel (because none of them are equal). The second equation was already multiplied by $-\text{4}$. We need to change one of the coefficients in the first equation so that we can cancel terms.

Specifically, we need to multiply $-\text{4}$ to get $-\text{20}$ so we can cancel the $a$-terms, or multiply the $-\text{4}$ to get $-\text{12}$ so we can cancel the $b$-terms.

For these equations, both options work. The coefficients of the $a$-terms and the $b$-terms in the second equation are multiples of the coefficients in the other equation. If we multiply by $\text{3}$ we can cancel the $b$-terms. But we could also multiply by $\text{5}$ which would allow us to cancel the $a$-terms. Both are good choices. Here we will multiply the first equation by $\text{3}$.

The coefficients we want to cancel are equal. So eliminating the terms requires subtracting both sides of the equation:

The correct answer can be any of these numbers: $\text{3}$, $-\text{3}$, $\text{5}$ or $-\text{5}$.

Remember that the goal of elimination is to cancel either the $k$-terms or the $p$-terms. So you need to change one of the coefficients in the first equations so that you can cancel terms.

For this question we need to multiply the first equation by a number so that we can solve the equations by elimination. In the original equations, none of the coefficients were ready to cancel (because none of them are equal). The second equation was already multiplied by $\text{2}$. We need to change one of the coefficients in the first equation so that we can cancel terms.

Specifically, we need to multiply $\text{4}$ to get $\text{6}$ so we can cancel the $k$-terms, or multiply the $\text{2}$ to get $\text{6}$ so we can cancel the $p$-terms.

The numbers in these equations point us toward cancelling the $p$-terms. This is because $\text{6}$ is a multiple of $\text{2}$. So if we multiply the first equation by $\text{3}$ the the coefficients will be equal.

The coefficients we want to cancel are equal. So eliminating the terms requires subtracting both sides of the equation:

The correct answer can be either of these numbers: $\text{3}$ or $-\text{3}$.

You need to solve these equations using the substitution method. So the first thing to do is pick one of the variables to isolate.

The question tells us to solve the equations using the substitution method. That is not necessarily the best method, but we do not have a choice!

None of the variables are isolated, so we will need to isolate one of them for the substitution step. It would be convenient if there was a coefficient of $\text{1}$ or $-\text{1}$, because that would make our work easier. But that is not the case. So we will just make $x$ the subject of the first equation:

Unfortunately, this equation has $\text{1}$ fraction in it. That is ugly, but it will work.

Substitute the expression for $x$ into the second equation. Then simplify and solve for the value of $y$:

Substitute this value of $y$ back into one of the equations and solve for $x$. In this case, the first equation has only $\text{1}$ negative sign while the other equation has $\text{3}$ negatives. So we will use the first equation.

The final answers are $x = \dfrac{34}{11}$ and $y = - \dfrac{60}{11}$.

You need to solve these equations using the substitution method. So the first thing to do is pick one of the variables to isolate.

The question tells us to solve the equations using the substitution method. That is not necessarily the best method, but we do not have a choice!

None of the variables are isolated, so we will need to isolate one of them for the substitution step. It would be convenient if there was a coefficient of $\text{1}$ or $-\text{1}$, because that would make our work easier. But that is not the case. So we will just make $x$ the subject of the first equation:

Unfortunately, this equation has $\text{2}$ fractions in it. That is ugly, but it will work.

Substitute the expression for $x$ into the second equation. Then simplify and solve for the value of $y$:

Substitute this value of $y$ back into one of the equations and solve for $x$. In this case, the second equation is better because it has fewer negatives than the first equation.

The final answers are $x = \dfrac{3}{5}$ and $y = - \dfrac{7}{5}$.

You need to solve these equations using the substitution method. So the first thing to do is pick one of the variables to isolate.

The question tells us to solve the equations using the substitution method. That is not necessarily the best method, but we do not have a choice!

None of the variables are isolated, so we will need to isolate one of them for the substitution step. It would be convenient if there was a coefficient of $\text{1}$ or $-\text{1}$, because that would make our work easier. But that is not the case. So we will just make $x$ the subject of the first equation:

Substitute the expression for $x$ into the second equation. Then simplify and solve for the value of $y$:

Substitute this value of $y$ back into one of the equations and solve for $x$. In this case, the first equation has only $\text{1}$ negative sign while the other equation has $\text{2}$ negatives. So we will use the first equation.

The final answers are $x = \dfrac{3}{4}$ and $y = - \dfrac{9}{8}$.

The best choice for substitution is based on which variable is easiest to isolate. In this case, you should look for the variable which is already isolated.

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is substitution. The substitution method works by using one of the equations to replace one of the variables in the other equation. We need to figure out which variable is the best choice to substitute for these two equations.

In these two equations, there are two $x$-terms and two $y$-terms:

The goal of substitution is to use one of the equations to remove a variable from the other equation.

In this case, the best choice is to substitute the $y$ from the second equation into the first equation. This is because in the second equation the $y$ is already isolated. To use substitution, we need a variable which is isolated (alone on one side of the equation).

In this case, the substitution looks like this:

Combining these equations by substitution, we get:

Since the second equation tells us that $y$ is equal to $y=2 x + 2$, we can substitute it into the other equation in place of $y$ straight away.

The best choice is to substitute the $y$ from the second equation into the first equation because it is already isolated.

The best choice for substitution is based on which variable is easiest to isolate. In this case, you should look for the variable which is already isolated.

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is substitution. The substitution method works by using one of the equations to replace one of the variables in the other equation. We need to figure out which variable is the best choice to substitute for these two equations.

In these two equations, there are two $x$-terms and two $y$-terms:

The goal of substitution is to use one of the equations to remove a variable from the other equation.

In this case, the best choice is to substitute the $y$ from the first equation into the second equation. This is because in the first equation the $y$ is already isolated. To use substitution, we need a variable which is isolated (alone on one side of the equation).

In this case, the substitution looks like this:

Combining these equations by substitution, we get:

Since the first equation tells us that $y$ is equal to $y=- 3 x - 4$, we can substitute it into the other equation in place of $y$ straight away.

The best choice is to substitute the $y$ from the first equation into the second equation because it is already isolated.

The best choice for substitution is based on which variable is easiest to isolate. In this case, you should look for the variable which is already isolated.

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is substitution. The substitution method works by using one of the equations to replace one of the variables in the other equation. We need to figure out which variable is the best choice to substitute for these two equations.

In these two equations, there are two $x$-terms and two $y$-terms:

The goal of substitution is to use one of the equations to remove a variable from the other equation.

In this case, the best choice is to substitute the $x$ from the first equation into the second equation. This is because in the first equation the $x$ is already isolated. To use substitution, we need a variable which is isolated (alone on one side of the equation).

In this case, the substitution looks like this:

Combining these equations by substitution, we get:

Since the first equation tells us that $x$ is equal to $x=- 5 y + 5$, we can substitute it into the other equation in place of $x$ straight away.

The best choice is to substitute the $x$ from the first equation into the second equation because it is already isolated.

The first thing you need to do (if you will use substitution to solve the pair of equations) is isolate one of the variables from one of the equations; then substitute what you get into the other equation.

When we solve simultaneous equations, we are working to find values for $y$ and $x$ which make both equations true!

We can solve these equations by substitution. Substitution requires isolating a variable. We need to look at the equations to decide which variable to isolate. We can see that the first equation looks good for this, because the $y$ has a coefficient of one:

Isolate $y$ in the first equation, which can be done in one step:

Now we can substitute the expression $5 x + 9$ into the second equation and solve:

Right now we have only half of the answer! To get the other half, we need to find the value of $y$. We can use the equation we got when we isolated $y$: it is the most convenient choice because $y$ is the subject of the equation.

The correct answers are $x = 0$ and $y = 9$.

The first thing you need to do (if you will use substitution to solve the pair of equations) is isolate one of the variables from one of the equations; then substitute what you get into the other equation.

When we solve simultaneous equations, we are working to find values for $x$ and $y$ which make both equations true!

We can solve these equations by substitution. Substitution requires isolating a variable. We need to look at the equations to decide which variable to isolate. We can see that the first equation looks good for this, because the $x$ has a coefficient of one:

Isolate $x$ in the first equation, which can be done in one step:

Now we can substitute the expression $- 3 y + 17$ into the second equation and solve:

Right now we have only half of the answer! To get the other half, we need to find the value of $x$. We can use the equation we got when we isolated $x$: it is the most convenient choice because $x$ is the subject of the equation.

The correct answers are $x = 2$ and $y = 5$.

The first thing you need to do (if you will use substitution to solve the pair of equations) is isolate one of the variables from one of the equations; then substitute what you get into the other equation.

When we solve simultaneous equations, we are working to find values for $y$ and $x$ which make both equations true!

We can solve these equations by substitution. Substitution requires isolating a variable. We need to look at the equations to decide which variable to isolate. We can see that the second equation looks good for this, because the $y$ has a coefficient of one:

Isolate $y$ in the second equation, which can be done in one step:

Now we can substitute the expression $- 6 x - 32$ into the first equation and solve:

Right now we have only half of the answer! To get the other half, we need to find the value of $y$. We can use the equation we got when we isolated $y$: it is the most convenient choice because $y$ is the subject of the equation.

The correct answers are $x = -5$ and $y = -2$.

You need to choose variables to represent the things you want to find and write equations based on the information in the question.

The two things we want to know in this question are the prices for each of the items (a pizza and a milkshake). To begin, we can pick a variable for each of these numbers. It is helpful to pick variables which remind you about the things in the question:

Next we need to write equations based on what the question tells us. In other words, we need to translate the words in the question into equations.

The first point says that "the total cost for the $\text{7}$ pizzas and $\text{5}$ milkshakes is $\text{R}\,\text{342}$." We can use the expression $7 p$ to represent the price of the $\text{7}$ pizzas. Similarly, the expression $5 m$ represents the price of the $\text{5}$ milkshakes. With these values we can write a full equation for the prices:

Now we can use the second point. It says, "a pizza costs $\text{R}\,\text{6}$ more than a milkshake." We can write this as an equation like this:

Now that we have two equations, we need to solve them simultaneously.

We could use elimination, but substitution is a better choice (because $p$ is already the subject). Substitute the second equation into the first equation and solve!

This means that the price of one milkshake is $\text{R}\,\text{25}$.

Finally, use the value we found for $m$ to find the value of $p$ (the price of a pizza). We can use either equation to do this, but the second one is easier to use (because $p$ is already isolated).

The price for one pizza is $\text{R}\,\text{31}$.

It is important to write the answer to a word problem as a complete sentence.

The price of the pizza is $\text{R}\,\text{31}$ while a milkshake costs $\text{R}\,\text{25}$.

You need to choose variables to represent the things you want to find and write equations based on the information in the question.

The two things we want to know in this question are the prices for each of the items (a wrap and a milkshake). To begin, we can pick a variable for each of these numbers. It is helpful to pick variables which remind you about the things in the question:

Next we need to write equations based on what the question tells us. In other words, we need to translate the words in the question into equations.

The first point says that "the total cost for the $\text{5}$ wraps and $\text{4}$ milkshakes is $\text{R}\,\text{228}$." We can use the expression $5 w$ to represent the price of the $\text{5}$ wraps. Similarly, the expression $4 m$ represents the price of the $\text{4}$ milkshakes. With these values we can write a full equation for the prices:

Now we can use the second point. It says, "a wrap costs $\text{R}\,\text{6}$ more than a milkshake." We can write this as an equation like this:

Now that we have two equations, we need to solve them simultaneously.

We could use elimination, but substitution is a better choice (because $w$ is already the subject). Substitute the second equation into the first equation and solve!

This means that the price of one milkshake is $\text{R}\,\text{22}$.

Finally, use the value we found for $m$ to find the value of $w$ (the price of a wrap). We can use either equation to do this, but the second one is easier to use (because $w$ is already isolated).

The price for one wrap is $\text{R}\,\text{28}$.

It is important to write the answer to a word problem as a complete sentence.

The price of the wrap is $\text{R}\,\text{28}$ while a milkshake costs $\text{R}\,\text{22}$.

You need to choose variables to represent the things you want to find and write equations based on the information in the question.

The two things we want to know in this question are the prices for each of the items (a salad and a hamburger). To begin, we can pick a variable for each of these numbers. It is helpful to pick variables which remind you about the things in the question:

Next we need to write equations based on what the question tells us. In other words, we need to translate the words in the question into equations.

The first point says that "the total cost for the $\text{4}$ salads and $\text{6}$ hamburgers is $\text{R}\,\text{298}$." We can use the expression $4 s$ to represent the price of the $\text{4}$ salads. Similarly, the expression $6 h$ represents the price of the $\text{6}$ hamburgers. With these values we can write a full equation for the prices:

Now we can use the second point. It says, "a salad costs $\text{R}\,\text{7}$ more than a hamburger." We can write this as an equation like this:

Now that we have two equations, we need to solve them simultaneously.

We could use elimination, but substitution is a better choice (because $s$ is already the subject). Substitute the second equation into the first equation and solve!

This means that the price of one hamburger is $\text{R}\,\text{27}$.

Finally, use the value we found for $h$ to find the value of $s$ (the price of a salad). We can use either equation to do this, but the second one is easier to use (because $s$ is already isolated).

The price for one salad is $\text{R}\,\text{34}$.

It is important to write the answer to a word problem as a complete sentence.

The price of the salad is $\text{R}\,\text{34}$ while a hamburger costs $\text{R}\,\text{27}$.

Notice that the $x$-coefficients are opposites. You should start by adding the equations together, which will cancel those terms out.

We have two equations and we need to solve them 'simultaneously'. That means we need to find two numbers, for $x$ and for $y$, which solve both of the equations.

There are two methods for solving equations simultaneously: elimination and substitution. The equations in this question are perfect for elimination because the $x$-coefficients are equal and opposite. If we add the equations together, the $x$-terms will cancel. (We could solve these equations using the substitution method but elimination will be faster.)

So we add the equations together. This means we add together the left sides of the equations and also the right sides of the equations.

Now we have an equation with only one variable, and we can solve it.

So $y = 6$. But we are not done yet: we also need an answer for $x$. We can get this using the answer we just got for $y$. Substitute this into either of the equations to get the answer - we can use either equation because we are looking for the number that solves both of them. In this case, neither of the equations is much more friendly than the other, so we will just use the first equation.

Now we have the complete answer: the numbers which solve the equations are $x = 4$ and $y = 6$. We can see this represented beautifully on a graph of the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are $x = 4$ and $y = 6$.

Notice that the $x$-coefficients are opposites. You should start by adding the equations together, which will cancel those terms out.

We have two equations and we need to solve them 'simultaneously'. That means we need to find two numbers, for $x$ and for $y$, which solve both of the equations.

There are two methods for solving equations simultaneously: elimination and substitution. The equations in this question are perfect for elimination because the $x$-coefficients are equal and opposite. If we add the equations together, the $x$-terms will cancel. (We could solve these equations using the substitution method but elimination will be faster.)

So we add the equations together. This means we add together the left sides of the equations and also the right sides of the equations.

Now we have an equation with only one variable, and we can solve it.

So $y = 6$. But we are not done yet: we also need an answer for $x$. We can get this using the answer we just got for $y$. Substitute this into either of the equations to get the answer - we can use either equation because we are looking for the number that solves both of them. In this case, neither of the equations is much more friendly than the other, so we will just use the first equation.

Now we have the complete answer: the numbers which solve the equations are $x = 4$ and $y = 6$. We can see this represented beautifully on a graph of the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are $x = 4$ and $y = 6$.

Notice that the $y$-coefficients are opposites. You should start by adding the equations together, which will cancel those terms out.

We have two equations and we need to solve them 'simultaneously'. That means we need to find two numbers, for $x$ and for $y$, which solve both of the equations.

There are two methods for solving equations simultaneously: elimination and substitution. The equations in this question are perfect for elimination because the $y$-coefficients are equal and opposite. If we add the equations together, the $y$-terms will cancel. (We could solve these equations using the substitution method but elimination will be faster.)

So we add the equations together. This means we add together the left sides of the equations and also the right sides of the equations.

Now we have an equation with only one variable, and we can solve it.

So $x = -2$. But we are not done yet: we also need an answer for $y$. We can get this using the answer we just got for $x$. Substitute this into either of the equations to get the answer - we can use either equation because we are looking for the number that solves both of them. In this case, neither of the equations is much more friendly than the other, so we will just use the first equation.

Now we have the complete answer: the numbers which solve the equations are $x = -2$ and $y = -1$. We can see this represented beautifully on a graph of the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are $x = -2$ and $y = -1$.

Read through the question carefully and underline the important information. Then you will need to pick variables to represent each of the unknown facts from the question: the two marks. Use these variables to write down two equations which summarize the information in the question.

We need to figure out the marks the students got on their tests. The first thing to do is to pick variables for each student's mark. It is helpful to pick variables which match the information we want, for example:

Now we can write equations with those variables. There are two pieces of information we have about the marks, which lead to two equations:

We can solve these equations simultaneously. Substitute the first equation into the second equation and solve. (You can solve these equations using elimination if you prefer.)

This means that Melanie's mark is $\text{65}$.

Now we can use Melanie's mark and one of the equations we have to find Abdulai's mark. The easiest way to do that is to substitute Melanie's mark back into the first equation:

The students achieved these marks: Abdulai earned $\text{84}$ and Melanie earned $\text{65}$.

Read through the question carefully and underline the important information. Then you will need to pick variables to represent each of the unknown facts from the question: the two marks. Use these variables to write down two equations which summarize the information in the question.

We need to figure out the marks the students got on their tests. The first thing to do is to pick variables for each student's mark. It is helpful to pick variables which match the information we want, for example:

Now we can write equations with those variables. There are two pieces of information we have about the marks, which lead to two equations:

We can solve these equations simultaneously. Substitute the first equation into the second equation and solve. (You can solve these equations using elimination if you prefer.)

This means that Mathe's mark is $\text{53}$.

Now we can use Mathe's mark and one of the equations we have to find Realeboha's mark. The easiest way to do that is to substitute Mathe's mark back into the first equation:

The students achieved these marks: Realeboha earned $\text{73}$ and Mathe earned $\text{53}$.

Read through the question carefully and underline the important information. Then you will need to pick variables to represent each of the unknown facts from the question: the two marks. Use these variables to write down two equations which summarize the information in the question.

We need to figure out the marks the students got on their tests. The first thing to do is to pick variables for each student's mark. It is helpful to pick variables which match the information we want, for example:

Now we can write equations with those variables. There are two pieces of information we have about the marks, which lead to two equations:

We can solve these equations simultaneously. Substitute the first equation into the second equation and solve. (You can solve these equations using elimination if you prefer.)

This means that Johan's mark is $\text{71}$.

Now we can use Johan's mark and one of the equations we have to find Yengwayo's mark. The easiest way to do that is to substitute Johan's mark back into the first equation:

The students achieved these marks: Yengwayo earned $\text{85}$ and Johan earned $\text{71}$.

This question is about two unknown numbers. And we need to find one of them (the smaller one). We know certain things about these numbers: they are positive, they are consecutive even integers, and they have a sum of $\text{38}$. We can solve this question using simultaneous equations.

The first thing to do is pick variables to represent the two numbers. Then we can write equations using those variables. It is usually helpful to pick variables which represent things we want to find. In this case, we are looking for two numbers, so these are good choices:

The first fact given in the question says that the numbers are "consecutive even integers". So both of the numbers are even, and they come one after another. For example, $\text{10}$ and $\text{12}$ are consecutive even numbers. Since we defined $n_{1}$ as the smaller number, $n_{2}$ must be $\text{2}$ more than $n_{1}$.

The "$\text{+2}$" skips the odd integer which sits between $n_{1}$ and $n_{2}$.

The second fact about the numbers tells us that "the sum of the numbers is $\text{38}$". Remember that sum means addition. So:

We can now use these two equations to find the answer to the question. But remember that we only need to find the smaller number (we do not need both of them). That means we need to find the value of $n_{1}$.

We can do this using substitution. If we substitute $n_2 = n_1 + 2$ into the equation $n_1 + n_2 = 38$, the $n_2$ terms will disappear, leaving $n_1$. Then we can solve for $n_1$, which is exactly what we want.

The result is $n_{1} = 18$. Notice that this means that the other number, $n_{2}$, must be $\text{20}$, because $n_{2} = n_{1} + 2$. This is perfect, because we know that the sum of the numbers is $\text{38}$, and $18 + 20 = 38$.

The smaller number is $\text{18}$.

This question is about two unknown numbers. And we need to find one of them (the larger one). We know certain things about these numbers: they are positive, they are consecutive even integers, and they have a sum of $\text{34}$. We can solve this question using simultaneous equations.

The first thing to do is pick variables to represent the two numbers. Then we can write equations using those variables. It is usually helpful to pick variables which represent things we want to find. In this case, we are looking for two numbers, so these are good choices:

The first fact given in the question says that the numbers are "consecutive even integers". So both of the numbers are even, and they come one after another. For example, $\text{10}$ and $\text{12}$ are consecutive even numbers. Since we defined $n_{1}$ as the smaller number, $n_{2}$ must be $\text{2}$ more than $n_{1}$.

The "$\text{+2}$" skips the odd integer which sits between $n_{1}$ and $n_{2}$.

The second fact about the numbers tells us that "the sum of the numbers is $\text{34}$". Remember that sum means addition. So:

We can now use these two equations to find the answer to the question. But remember that we only need to find the larger number (we do not need both of them). That means we need to find the value of $n_{2}$.

We can solve this using substitution. However, remember that we want the value of $n_2$ (the larger number). We can start by rearranging the first equation to make $n_1$ the subject. Then the substitution step will remove $n_1$ from the second equation and we can solve for $n_2$. (This is not required - it just makes the solution faster.)

Now substitute $n_2 - 2$ into the other equation and solve for $n_2$.

The result is $n_{2} = 18$. Notice that this means that the other number, $n_{1}$, must be $\text{16}$, because $n_{2} = n_{1} + 2$. This is perfect, because we know that the sum of the numbers is $\text{34}$, and $18 + 16 = 34$.

The larger number is $\text{18}$.

This question is about two unknown numbers. And we need to find one of them (the smaller one). We know certain things about these numbers: they are positive, they are consecutive odd integers, and they have a sum of $\text{32}$. We can solve this question using simultaneous equations.

The first thing to do is pick variables to represent the two numbers. Then we can write equations using those variables. It is usually helpful to pick variables which represent things we want to find. In this case, we are looking for two numbers, so these are good choices:

The first fact given in the question says that the numbers are "consecutive odd integers". So both of the numbers are odd, and they come one after another. For example, $\text{11}$ and $\text{13}$ are consecutive odd numbers. Since we defined $n_{1}$ as the smaller number, $n_{2}$ must be $\text{2}$ more than $n_{1}$.

The "$\text{+2}$" skips the even integer which sits between $n_{1}$ and $n_{2}$.

The second fact about the numbers tells us that "the sum of the numbers is $\text{32}$". Remember that sum means addition. So:

We can now use these two equations to find the answer to the question. But remember that we only need to find the smaller number (we do not need both of them). That means we need to find the value of $n_{1}$.

We can do this using substitution. If we substitute $n_2 = n_1 + 2$ into the equation $n_1 + n_2 = 32$, the $n_2$ terms will disappear, leaving $n_1$. Then we can solve for $n_1$, which is exactly what we want.

The result is $n_{1} = 15$. Notice that this means that the other number, $n_{2}$, must be $\text{17}$, because $n_{2} = n_{1} + 2$. This is perfect, because we know that the sum of the numbers is $\text{32}$, and $15 + 17 = 32$.

The smaller number is $\text{15}$.

You need to find Nkiru's age based on the given information. The first thing to do is define variables for the two ages, which are unknown. Then write equations with those variables based on the facts in the question.

We need to determine Nkiru's age based on the information given in the question. To do this, we can write equations to represent the information, and then solve the equations.

The first thing to do is define variables for the unknowns in the question. The unknown values are the ages of both Adaeze and Nkiru. We can use any variables we want, but it is best to choose variables which represent the information we want. In this case, we want ages, and we need to distinguish them somehow. Here is one good pair of options:

Now let's write equations using these variables. From the question we know that: "Nkiru is $\text{12}$ years younger than her sister, Adaeze." We need to translate that into mathematics. The key word is younger, which tells us to use subtraction to relate the ages.

We also know that: "In $\text{10}$ years, Adaeze will be $\text{7}$ times as old as Nkiru." Now we are looking into the future and we need to represent that information in maths. Using the variables we already have, we can write:

These are the ages at which Adaeze will be $\text{7}$ times as old as Nkiru. We can put all this together as follows:

Now we have to solve two equations, both of them including the variables $a_{A}$ and $a_{N}$. The easiest way to solve them is substitution (you can use elimination if you prefer). The first equation is $a_N = a_A - 12$. We can substitute this into the second equation and solve for $a_{A}$.

Terrific: this means that Adaeze is $\text{4}$ years old. But the question asked for us to find Nkiru's age. We can find it using the first equation, which relates the two ages:

So we finally got the answer to the question. But wait a minute: $a_N$ represents the age of a person. It can't be negative! This either means that we made a mistake, or that there is no solution to the question. There is no mistake in the work: it turns out that the facts given about the two people's ages are not possible! The numbers agree with all the relationships given in the question, but we cannot forget that the numbers in this question have meaning. They refer to how many years it has been since someone was born. And that cannot be a negative number.

The correct answer is: no solution.

You need to find Talatu's age based on the given information. The first thing to do is define variables for the two ages, which are unknown. Then write equations with those variables based on the facts in the question.

We need to determine Talatu's age based on the information given in the question. To do this, we can write equations to represent the information, and then solve the equations.

The first thing to do is define variables for the unknowns in the question. The unknown values are the ages of both Adedapo and Talatu. We can use any variables we want, but it is best to choose variables which represent the information we want. In this case, we want ages, and we need to distinguish them somehow. Here is one good pair of options:

Now let's write equations using these variables. From the question we know that: "Adedapo is $\text{18}$ years older than his sister, Talatu." We need to translate that into mathematics. The key word is older, which tells us to use addition to relate the ages.

We also know that: "In $\text{8}$ years, Adedapo will be $\text{3}$ times as old as Talatu." Now we are looking into the future and we need to represent that information in maths. Using the variables we already have, we can write:

These are the ages at which Adedapo will be $\text{3}$ times as old as Talatu. We can put all this together as follows:

Now we have to solve two equations, both of them including the variables $a_{A}$ and $a_{T}$. The easiest way to solve them is substitution (you can use elimination if you prefer). The first equation is $a_A = a_T + 18$. We can substitute this into the second equation and solve for $a_{T}$.

Terrific: we have the answer.

Talatu is $\text{1}$ year old.

The correct answer is: 1.

You need to find Justine's age based on the given information. The first thing to do is define variables for the two ages, which are unknown. Then write equations with those variables based on the facts in the question.

We need to determine Justine's age based on the information given in the question. To do this, we can write equations to represent the information, and then solve the equations.

The first thing to do is define variables for the unknowns in the question. The unknown values are the ages of both Kelly and Justine. We can use any variables we want, but it is best to choose variables which represent the information we want. In this case, we want ages, and we need to distinguish them somehow. Here is one good pair of options:

Now let's write equations using these variables. From the question we know that: "Kelly is $\text{9}$ years older than her sister, Justine." We need to translate that into mathematics. The key word is older, which tells us to use addition to relate the ages.

We also know that: "In $\text{5}$ years, Kelly will be $\text{2}$ times as old as Justine." Now we are looking into the future and we need to represent that information in maths. Using the variables we already have, we can write:

These are the ages at which Kelly will be $\text{2}$ times as old as Justine. We can put all this together as follows:

Now we have to solve two equations, both of them including the variables $a_{K}$ and $a_{J}$. The easiest way to solve them is substitution (you can use elimination if you prefer). The first equation is $a_K = a_J + 9$. We can substitute this into the second equation and solve for $a_{J}$.

Terrific: we have the answer.

Justine is $\text{4}$ years old.

The correct answer is: 4.

You can solve the equations simultaneously using elimination or substitution. The first thing you should do is decide which method is the better choice for these equations.

To solve the given equations simultaneously, we can use either substitution or elimination. The best choice depends on the arrangement of the equations. Since the $x$ is isolated in the second equation, the best method here is substitution. We can substitute $- 4 y + 12$ into the first equation in place of $x$.

Substitute the $x$ and solve for $y$:

Super - we know that $y = 2$.

Now we can use the $y$-value to find the value of $x$. We can do this using either of the equations in the question. If one of the equations is simpler or more convenient (for example, if it has an isolated variable or has fewer negative signs) it is better to choose that equation. In this case, the second equation will be better, because $x$ is already isolated.

The answer is the pair of numbers $x = 4$ and $y = 2$. On the Cartesian plane, this looks like this:

The values which solve these two equations are $x = 4$ and $y = 2$.

You can solve the equations simultaneously using elimination or substitution. The first thing you should do is decide which method is the better choice for these equations.

To solve the given equations simultaneously, we can use either substitution or elimination. The best choice depends on the arrangement of the equations. In this case we need to modify the equations before we can do elimination or substitution. And we only need to make a small change to arrange the equations for substitution: we just need to subtract the $20$ term to the other side of the first equation.

First rearrange the first equation to set up the substitution:

Now do the substitution and complete the solution to find $x$.

Super - we know that $x = 4$.

Now we can use the $x$-value to find the value of $y$. We can do this using either of the equations in the question. If one of the equations is simpler or more convenient (for example, if it has an isolated variable or has fewer negative signs) it is better to choose that equation. In this case, the second equation is better because it has fewer negative signs.

The answer is the pair of numbers $x = 4$ and $y = -8$. On the Cartesian plane, this looks like this:

The values which solve these two equations are $x = 4$ and $y = -8$.

You can solve the equations simultaneously using elimination or substitution. The first thing you should do is decide which method is the better choice for these equations.

To solve the given equations simultaneously, we can use either substitution or elimination. The best choice depends on the arrangement of the equations. These equations are not arranged nicely for elimination or substitution. However, notice that the coefficients in the second equation are all a multiple of $-\text{3}$. So we can divide the equation by $-\text{3}$ to isolate $x$ without picking up any fractions!

First rearrange the second equation to set up the substitution:

Now do the substitution and complete the solution to find $y$.

Super - we know that $y = 1$.

Now we can use the $y$-value to find the value of $x$. We can do this using either of the equations in the question. If one of the equations is simpler or more convenient (for example, if it has an isolated variable or has fewer negative signs) it is better to choose that equation. In this case, the second equation is better because it has fewer negative signs.

The answer is the pair of numbers $x = -1$ and $y = 1$. On the Cartesian plane, this looks like this:

The values which solve these two equations are $x = -1$ and $y = 1$.

Start by choosing variables for the things you want to know. Then write equations with those variables to summarise the information in the question.

In this question we want to find the age of two people: Dumile and his son, Anathi. We can solve this by setting up two equations and solving them simultaneously.

Start by choosing variables to represent the ages of the father and the son. (We need to do this because we don't know the ages!) It is a good idea to choose variables that match what we are describing. So:

Now we want to use those variables to write equations. The first piece of information from the question tells us that "Dumile is now $\text{11}$ times as old as Anathi." As an equation, this is:

The second piece of information says that in "$\text{5}$ years... Dumile will be $\text{6}$ times as old as his son." In $\text{5}$ years Dumile will be $d + \text{5}$ years old, and similarly Anathi will be $a + \text{5}$ years old. Then:

Now we have a pair of simultaneous equations! Substitute the first equation into the second equation and solve. (You can solve the equations using elimination if you prefer. In that case, the easiest option is to subtract the equations to cancel $d$.)

Great! Now we know that Anathi is $\text{5}$ years old.

We can now find Dumile's age. Substitute Anathi's age into one of the equations to do this. In this case, the first equation is the simpler choice for this calculation.

Write your final answer: Dumile is $\text{55}$ years old and Anathi is $\text{5}$ years old.

Start by choosing variables for the things you want to know. Then write equations with those variables to summarise the information in the question.

In this question we want to find the age of two people: Phetoho and his son, Azubuike. We can solve this by setting up two equations and solving them simultaneously.

Start by choosing variables to represent the ages of the father and the son. (We need to do this because we don't know the ages!) It is a good idea to choose variables that match what we are describing. So:

Now we want to use those variables to write equations. The first piece of information from the question tells us that "Phetoho is now $\text{6}$ times as old as Azubuike." As an equation, this is:

The second piece of information says that in "$\text{9}$ years... Phetoho will be $\text{3}$ times as old as his son." In $\text{9}$ years Phetoho will be $p + \text{9}$ years old, and similarly Azubuike will be $a + \text{9}$ years old. Then:

Now we have a pair of simultaneous equations! Substitute the first equation into the second equation and solve. (You can solve the equations using elimination if you prefer. In that case, the easiest option is to subtract the equations to cancel $p$.)

Great! Now we know that Azubuike is $\text{6}$ years old.

We can now find Phetoho's age. Substitute Azubuike's age into one of the equations to do this. In this case, the first equation is the simpler choice for this calculation.

Write your final answer: Phetoho is $\text{36}$ years old and Azubuike is $\text{6}$ years old.

Start by choosing variables for the things you want to know. Then write equations with those variables to summarise the information in the question.

In this question we want to find the age of two people: Chukwudi and his son, Dumile. We can solve this by setting up two equations and solving them simultaneously.

Start by choosing variables to represent the ages of the father and the son. (We need to do this because we don't know the ages!) It is a good idea to choose variables that match what we are describing. So:

Now we want to use those variables to write equations. The first piece of information from the question tells us that "Chukwudi is now $\text{5}$ times as old as Dumile." As an equation, this is:

The second piece of information says that in "$\text{5}$ years... Chukwudi will be $\text{3}$ times as old as his son." In $\text{5}$ years Chukwudi will be $c + \text{5}$ years old, and similarly Dumile will be $d + \text{5}$ years old. Then:

Now we have a pair of simultaneous equations! Substitute the first equation into the second equation and solve. (You can solve the equations using elimination if you prefer. In that case, the easiest option is to subtract the equations to cancel $c$.)

Great! Now we know that Dumile is $\text{5}$ years old.

We can now find Chukwudi's age. Substitute Dumile's age into one of the equations to do this. In this case, the first equation is the simpler choice for this calculation.

Write your final answer: Chukwudi is $\text{25}$ years old and Dumile is $\text{5}$ years old.

The first thing you need to do is substitute $y=- 3 x + 2$ into the second equation.

We must solve the two equations 'simultaneously'. This means we want one pair of $x$ and $y$ values which solves both equations.

There are two methods for solving equations simultaneously: elimination and substitution. For the equations in this question, substitution is the better choice. This is because the first equation already shows us that $y$ is equal to $y=- 3 x + 2$, which is exactly what we need for the substitution method. (We could solve these equations using the elimination method, but it will require more work.)

So we can substitute $- 3 x + 2$ into the second equation in place of $y$:

Now that there is only one variable in the equation, we can solve the equation. Distribute the $\text{2}$ and get on with solving the equation.

So $x = 2$. But remember that we also need an answer for $y$. It is important to remember that you can use either equation to calculate $y$, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easiest choice is the first equation, because it is arranged in a more useful way.

The numbers which solve the equations simultaneously are $x = 2$ and $y = -4$. We can see this represented beautifully if we graph the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are $x = 2$ and $y = -4$.

The first thing you need to do is substitute $y=- 4 x + 3$ into the first equation.

We must solve the two equations 'simultaneously'. This means we want one pair of $x$ and $y$ values which solves both equations.

There are two methods for solving equations simultaneously: elimination and substitution. For the equations in this question, substitution is the better choice. This is because the second equation already shows us that $y$ is equal to $y=- 4 x + 3$, which is exactly what we need for the substitution method. (We could solve these equations using the elimination method, but it will require more work.)

So we can substitute $- 4 x + 3$ into the first equation in place of $y$:

Now that there is only one variable in the equation, we can solve the equation. Distribute the $\text{3}$ and get on with solving the equation.

So $x = 1$. But remember that we also need an answer for $y$. It is important to remember that you can use either equation to calculate $y$, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easiest choice is the second equation, because it is arranged in a more useful way.

The numbers which solve the equations simultaneously are $x = 1$ and $y = -1$. We can see this represented beautifully if we graph the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are $x = 1$ and $y = -1$.

The first thing you need to do is substitute $x=2 y - 4$ into the first equation.

We must solve the two equations 'simultaneously'. This means we want one pair of $x$ and $y$ values which solves both equations.

There are two methods for solving equations simultaneously: elimination and substitution. For the equations in this question, substitution is the better choice. This is because the second equation already shows us that $x$ is equal to $x=2 y - 4$, which is exactly what we need for the substitution method. (We could solve these equations using the elimination method, but it will require more work.)

So we can substitute $2 y - 4$ into the first equation in place of $x$:

Now that there is only one variable in the equation, we can solve the equation. Distribute the $\text{5}$ and get on with solving the equation.

So $y = 0$. But remember that we also need an answer for $x$. It is important to remember that you can use either equation to calculate $x$, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easiest choice is the second equation, because it is arranged in a more useful way.

The numbers which solve the equations simultaneously are $x = -4$ and $y = 0$. We can see this represented beautifully if we graph the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are $x = -4$ and $y = 0$.

For the first equation, the key word is sum. For the second equation the key word is more. Use these key words to change the words into operations, for example, addition or multiplication.

In this question we need to translate words into equations. This can be challenging. One useful approach is to look for important words which tell us what numbers and calculations to use. Here are some common words and what they mean when we write mathematical expressions and equations:

With these key words in mind, let's identify the key parts/words in each of these facts. Then we can translate each of the parts into maths.

The question tells us that we should use the variable $a$ for Adaeze's mark and $b$ for Babalwe's mark. So we can break up the first fact like this:

The first fact is equivalent to this equation: $a + b = 133$.

Similarly, we can identify key parts of the second fact, and translate each into an expression.

This equation, $a = b + 11$, means that Adaeze's mark is more that Babalwe's mark.

The correct answers are:

For the first equation, the key word is sum. For the second equation the key word is more. Use these key words to change the words into operations, for example, addition or multiplication.

In this question we need to translate words into equations. This can be challenging. One useful approach is to look for important words which tell us what numbers and calculations to use. Here are some common words and what they mean when we write mathematical expressions and equations:

With these key words in mind, let's identify the key parts/words in each of these facts. Then we can translate each of the parts into maths.

The question tells us that we should use the variable $a$ for Adefoluke's mark and $b$ for Bukelwa's mark. So we can break up the first fact like this:

The first fact is equivalent to this equation: $a + b = 121$.

Similarly, we can identify key parts of the second fact, and translate each into an expression.

This equation, $a = b + 9$, means that Adefoluke's mark is more that Bukelwa's mark.

The correct answers are:

For the first equation, the key word is sum. For the second equation the key word is more. Use these key words to change the words into operations, for example, addition or multiplication.

In this question we need to translate words into equations. This can be challenging. One useful approach is to look for important words which tell us what numbers and calculations to use. Here are some common words and what they mean when we write mathematical expressions and equations:

With these key words in mind, let's identify the key parts/words in each of these facts. Then we can translate each of the parts into maths.

The question tells us that we should use the variable $a$ for Andrew's mark and $b$ for Babatunde's mark. So we can break up the first fact like this:

The first fact is equivalent to this equation: $a + b = 153$.

Similarly, we can identify key parts of the second fact, and translate each into an expression.

This equation, $a = b + 11$, means that Andrew's mark is more that Babatunde's mark.

The correct answers are: